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< body >
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< h1 class = "title" > Cubic configuration for the Stewart Platform< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org43e3f4a" > 1. Questions we wish to answer with this analysis< / a > < / li >
< li > < a href = "#org9890470" > 2. Configuration Analysis - Stiffness Matrix< / a >
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< ul >
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< li > < a href = "#org5d80bd3" > 2.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center< / a > < / li >
< li > < a href = "#orga5ac347" > 2.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center< / a > < / li >
< li > < a href = "#org47743ef" > 2.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center< / a > < / li >
< li > < a href = "#orgd0daf23" > 2.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center< / a > < / li >
< li > < a href = "#org6729b53" > 2.5. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org0969b06" > 3. Cubic size analysis< / a > < / li >
< li > < a href = "#org0f56064" > 4. initializeCubicConfiguration< / a >
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< ul >
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< li > < a href = "#orge3f31b8" > 4.1. Function description< / a > < / li >
< li > < a href = "#org2580fde" > 4.2. Optional Parameters< / a > < / li >
< li > < a href = "#org230a253" > 4.3. Cube Creation< / a > < / li >
< li > < a href = "#orgd1a04ef" > 4.4. Vectors of each leg< / a > < / li >
< li > < a href = "#orgfec0778" > 4.5. Verification of Height of the Stewart Platform< / a > < / li >
< li > < a href = "#orgc3cd1d1" > 4.6. Determinate the location of the joints< / a > < / li >
< li > < a href = "#org9063896" > 4.7. Returns Stewart Structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org5807740" > 5. Tests< / a >
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< ul >
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< li > < a href = "#org8809c47" > 5.1. First attempt to parametrisation< / a > < / li >
< li > < a href = "#org71c64bc" > 5.2. Second attempt< / a > < / li >
< li > < a href = "#orgc132128" > 5.3. Generate the Stewart platform for a Cubic configuration< / a > < / li >
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< / ul >
< / li >
< / ul >
< / div >
< / div >
< p >
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The discovery of the Cubic configuration is done in < a class = 'org-ref-reference' href = "#geng94_six_degree_of_freed_activ" > geng94_six_degree_of_freed_activ< / a > .
Further analysis is conducted in < a class = 'org-ref-reference' href = "#jafari03_orthog_gough_stewar_platf_microm" > jafari03_orthog_gough_stewar_platf_microm< / a > .
< / p >
< p >
People using orthogonal/cubic configuration: < a class = 'org-ref-reference' href = "#preumont07_six_axis_singl_stage_activ" > preumont07_six_axis_singl_stage_activ< / a > .
< / p >
< p >
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The specificity of the Cubic configuration is that each actuator is orthogonal with the others.
< / p >
< p >
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To generate and study the Cubic configuration, < code > initializeCubicConfiguration< / code > is used (description in section < a href = "#org7e73a4b" > 4< / a > ).
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< / p >
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< div id = "outline-container-org43e3f4a" class = "outline-2" >
< h2 id = "org43e3f4a" > < span class = "section-number-2" > 1< / span > Questions we wish to answer with this analysis< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< p >
The goal is to study the benefits of using a cubic configuration:
< / p >
< ul class = "org-ul" >
< li > Equal stiffness in all the degrees of freedom?< / li >
< li > No coupling between the actuators?< / li >
< li > Is the center of the cube an important point?< / li >
< / ul >
< / div >
< / div >
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< div id = "outline-container-org9890470" class = "outline-2" >
< h2 id = "org9890470" > < span class = "section-number-2" > 2< / span > Configuration Analysis - Stiffness Matrix< / h2 >
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< div class = "outline-text-2" id = "text-2" >
< / div >
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< div id = "outline-container-org5d80bd3" class = "outline-3" >
< h3 id = "org5d80bd3" > < span class = "section-number-3" > 2.1< / span > Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
< p >
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We create a cubic Stewart platform (figure < a href = "#org813c22d" > 1< / a > ) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot).
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The Jacobian matrix is estimated at the location of the center of the cube.
< / p >
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< div id = "org813c22d" class = "figure" >
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< p > < img src = "./figs/3d-cubic-stewart-aligned.png" alt = "3d-cubic-stewart-aligned.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Centered cubic configuration< / p >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'H_tot'< / span > , < span style = "color: #BFEBBF;" > 100< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Total height of the Hexapod [mm]< / span >
< span style = "color: #CC9393;" > 'L'< / span > , < span style = "color: #BFEBBF;" > 200< / span > < span style = "color: #7CB8BB;" > /< / span > sqrt< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Size of the Cube [mm]< / span >
< span style = "color: #CC9393;" > 'H'< / span > , < span style = "color: #BFEBBF;" > 60< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between base joints and platform joints [mm]< / span >
< span style = "color: #CC9393;" > 'H0'< / span > , < span style = "color: #BFEBBF;" > 200< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 60< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between the corner of the cube and the plane containing the base joints [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
stewart = initializeCubicConfiguration< span style = "color: #DCDCCC;" > (< / span > opts< span style = "color: #DCDCCC;" > )< / span > ;
opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'Jd_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 50< / span > < span style = "color: #BFEBBF;" > ]< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]< / span >
< span style = "color: #CC9393;" > 'Jf_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 50< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for force location from the top of the mobile platform [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
stewart = computeGeometricalProperties< span style = "color: #DCDCCC;" > (< / span > stewart, opts< span style = "color: #DCDCCC;" > )< / span > ;
save< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #CC9393;" > './mat/stewart.mat', 'stewart'< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K = stewart.Jd'< span style = "color: #7CB8BB;" > *< / span > stewart.Jd;
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< td class = "org-right" > 4.8e-18< / td >
< td class = "org-right" > 5.3e-19< / td >
< td class = "org-right" > -6.1e-20< / td >
< td class = "org-right" > 1e-18< / td >
< td class = "org-right" > 0.027< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
2019-03-26 09:25:04 +01:00
< div id = "outline-container-orga5ac347" class = "outline-3" >
< h3 id = "orga5ac347" > < span class = "section-number-3" > 2.2< / span > Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center< / h3 >
2019-03-25 18:12:43 +01:00
< div class = "outline-text-3" id = "text-2-2" >
< p >
2019-03-26 09:25:04 +01:00
We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure < a href = "#org813c22d" > 1< / a > ).
2019-03-25 18:12:43 +01:00
The Jacobian matrix is not estimated at the location of the center of the cube.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'H_tot'< / span > , < span style = "color: #BFEBBF;" > 100< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Total height of the Hexapod [mm]< / span >
< span style = "color: #CC9393;" > 'L'< / span > , < span style = "color: #BFEBBF;" > 200< / span > < span style = "color: #7CB8BB;" > /< / span > sqrt< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Size of the Cube [mm]< / span >
< span style = "color: #CC9393;" > 'H'< / span > , < span style = "color: #BFEBBF;" > 60< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between base joints and platform joints [mm]< / span >
< span style = "color: #CC9393;" > 'H0'< / span > , < span style = "color: #BFEBBF;" > 200< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 60< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between the corner of the cube and the plane containing the base joints [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
stewart = initializeCubicConfiguration< span style = "color: #DCDCCC;" > (< / span > opts< span style = "color: #DCDCCC;" > )< / span > ;
opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'Jd_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > ]< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]< / span >
< span style = "color: #CC9393;" > 'Jf_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for force location from the top of the mobile platform [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
stewart = computeGeometricalProperties< span style = "color: #DCDCCC;" > (< / span > stewart, opts< span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K = stewart.Jd'< span style = "color: #7CB8BB;" > *< / span > stewart.Jd;
< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 1.9e-18< / td >
< td class = "org-right" > -2.3e-17< / td >
< td class = "org-right" > 1.5e-18< / td >
< td class = "org-right" > -0.1< / td >
< td class = "org-right" > -1.5e-17< / td >
< / tr >
< tr >
< td class = "org-right" > 1.9e-18< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 6.8e-18< / td >
< td class = "org-right" > 0.1< / td >
< td class = "org-right" > -1.6e-18< / td >
< td class = "org-right" > 4.8e-18< / td >
< / tr >
< tr >
< td class = "org-right" > -2.3e-17< / td >
< td class = "org-right" > 6.8e-18< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > -5.1e-19< / td >
< td class = "org-right" > -5.5e-18< / td >
< td class = "org-right" > 5.3e-19< / td >
< / tr >
< tr >
< td class = "org-right" > 1.5e-18< / td >
< td class = "org-right" > 0.1< / td >
< td class = "org-right" > -5.1e-19< / td >
< td class = "org-right" > 0.012< / td >
< td class = "org-right" > -3e-19< / td >
< td class = "org-right" > 3.1e-19< / td >
< / tr >
< tr >
< td class = "org-right" > -0.1< / td >
< td class = "org-right" > -1.6e-18< / td >
< td class = "org-right" > -5.5e-18< / td >
< td class = "org-right" > -3e-19< / td >
< td class = "org-right" > 0.012< / td >
< td class = "org-right" > 1.9e-18< / td >
< / tr >
< tr >
< td class = "org-right" > -1.5e-17< / td >
< td class = "org-right" > 4.8e-18< / td >
< td class = "org-right" > 5.3e-19< / td >
< td class = "org-right" > 3.1e-19< / td >
< td class = "org-right" > 1.9e-18< / td >
< td class = "org-right" > 0.027< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
2019-03-26 09:25:04 +01:00
< div id = "outline-container-org47743ef" class = "outline-3" >
< h3 id = "org47743ef" > < span class = "section-number-3" > 2.3< / span > Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center< / h3 >
2019-03-25 18:12:43 +01:00
< div class = "outline-text-3" id = "text-2-3" >
< p >
2019-03-26 09:25:04 +01:00
Here, the "center" of the Stewart platform is not at the cube center (figure < a href = "#org8a2bd4c" > 2< / a > ).
2019-03-25 18:12:43 +01:00
The Jacobian is estimated at the cube center.
< / p >
2019-03-26 09:25:04 +01:00
< div id = "org8a2bd4c" class = "figure" >
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< p > < img src = "./figs/3d-cubic-stewart-misaligned.png" alt = "3d-cubic-stewart-misaligned.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Not centered cubic configuration< / p >
< / div >
< p >
The center of the cube is at \(z = 110\).
The Stewart platform is from \(z = H_0 = 75\) to \(z = H_0 + H_{tot} = 175\).
The center height of the Stewart platform is then at \(z = \frac{175-75}{2} = 50\).
The center of the cube from the top platform is at \(z = 110 - 175 = -65\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'H_tot'< / span > , < span style = "color: #BFEBBF;" > 100< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Total height of the Hexapod [mm]< / span >
< span style = "color: #CC9393;" > 'L'< / span > , < span style = "color: #BFEBBF;" > 220< / span > < span style = "color: #7CB8BB;" > /< / span > sqrt< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Size of the Cube [mm]< / span >
< span style = "color: #CC9393;" > 'H'< / span > , < span style = "color: #BFEBBF;" > 60< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between base joints and platform joints [mm]< / span >
< span style = "color: #CC9393;" > 'H0'< / span > , < span style = "color: #BFEBBF;" > 75< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between the corner of the cube and the plane containing the base joints [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
stewart = initializeCubicConfiguration< span style = "color: #DCDCCC;" > (< / span > opts< span style = "color: #DCDCCC;" > )< / span > ;
opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'Jd_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 65< / span > < span style = "color: #BFEBBF;" > ]< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]< / span >
< span style = "color: #CC9393;" > 'Jf_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 65< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for force location from the top of the mobile platform [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
stewart = computeGeometricalProperties< span style = "color: #DCDCCC;" > (< / span > stewart, opts< span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K = stewart.Jd'< span style = "color: #7CB8BB;" > *< / span > stewart.Jd;
< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-right" > 2< / td >
< td class = "org-right" > -1.8e-17< / td >
< td class = "org-right" > 2.6e-17< / td >
< td class = "org-right" > 3.3e-18< / td >
< td class = "org-right" > 0.04< / td >
< td class = "org-right" > 1.7e-19< / td >
< / tr >
< tr >
< td class = "org-right" > -1.8e-17< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 1.9e-16< / td >
< td class = "org-right" > -0.04< / td >
< td class = "org-right" > 2.2e-19< / td >
< td class = "org-right" > -5.3e-19< / td >
< / tr >
< tr >
< td class = "org-right" > 2.6e-17< / td >
< td class = "org-right" > 1.9e-16< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > -8.9e-18< / td >
< td class = "org-right" > 6.5e-19< / td >
< td class = "org-right" > -5.8e-19< / td >
< / tr >
< tr >
< td class = "org-right" > 3.3e-18< / td >
< td class = "org-right" > -0.04< / td >
< td class = "org-right" > -8.9e-18< / td >
< td class = "org-right" > 0.0089< / td >
< td class = "org-right" > -9.3e-20< / td >
< td class = "org-right" > 9.8e-20< / td >
< / tr >
< tr >
< td class = "org-right" > 0.04< / td >
< td class = "org-right" > 2.2e-19< / td >
< td class = "org-right" > 6.5e-19< / td >
< td class = "org-right" > -9.3e-20< / td >
< td class = "org-right" > 0.0089< / td >
< td class = "org-right" > -2.4e-18< / td >
< / tr >
< tr >
< td class = "org-right" > 1.7e-19< / td >
< td class = "org-right" > -5.3e-19< / td >
< td class = "org-right" > -5.8e-19< / td >
< td class = "org-right" > 9.8e-20< / td >
< td class = "org-right" > -2.4e-18< / td >
< td class = "org-right" > 0.032< / td >
< / tr >
< / tbody >
< / table >
< p >
We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiffness matrix is not diagonal.
< / p >
< / div >
< / div >
2019-03-26 09:25:04 +01:00
< div id = "outline-container-orgd0daf23" class = "outline-3" >
< h3 id = "orgd0daf23" > < span class = "section-number-3" > 2.4< / span > Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center< / h3 >
2019-03-25 18:12:43 +01:00
< div class = "outline-text-3" id = "text-2-4" >
< p >
Here, the "center" of the Stewart platform is not at the cube center.
The Jacobian is estimated at the center of the Stewart platform.
< / p >
< p >
The center of the cube is at \(z = 110\).
The Stewart platform is from \(z = H_0 = 75\) to \(z = H_0 + H_{tot} = 175\).
The center height of the Stewart platform is then at \(z = \frac{175-75}{2} = 50\).
The center of the cube from the top platform is at \(z = 110 - 175 = -65\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'H_tot'< / span > , < span style = "color: #BFEBBF;" > 100< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Total height of the Hexapod [mm]< / span >
< span style = "color: #CC9393;" > 'L'< / span > , < span style = "color: #BFEBBF;" > 220< / span > < span style = "color: #7CB8BB;" > /< / span > sqrt< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Size of the Cube [mm]< / span >
< span style = "color: #CC9393;" > 'H'< / span > , < span style = "color: #BFEBBF;" > 60< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between base joints and platform joints [mm]< / span >
< span style = "color: #CC9393;" > 'H0'< / span > , < span style = "color: #BFEBBF;" > 75< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between the corner of the cube and the plane containing the base joints [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
stewart = initializeCubicConfiguration< span style = "color: #DCDCCC;" > (< / span > opts< span style = "color: #DCDCCC;" > )< / span > ;
opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'Jd_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 60< / span > < span style = "color: #BFEBBF;" > ]< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]< / span >
< span style = "color: #CC9393;" > 'Jf_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 60< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for force location from the top of the mobile platform [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
stewart = computeGeometricalProperties< span style = "color: #DCDCCC;" > (< / span > stewart, opts< span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K = stewart.Jd'< span style = "color: #7CB8BB;" > *< / span > stewart.Jd;
< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-right" > 2< / td >
< td class = "org-right" > -1.8e-17< / td >
< td class = "org-right" > 2.6e-17< / td >
< td class = "org-right" > -5.7e-19< / td >
< td class = "org-right" > 0.03< / td >
< td class = "org-right" > 1.7e-19< / td >
< / tr >
< tr >
< td class = "org-right" > -1.8e-17< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 1.9e-16< / td >
< td class = "org-right" > -0.03< / td >
< td class = "org-right" > 2.2e-19< / td >
< td class = "org-right" > -5.3e-19< / td >
< / tr >
< tr >
< td class = "org-right" > 2.6e-17< / td >
< td class = "org-right" > 1.9e-16< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > -1.5e-17< / td >
< td class = "org-right" > 6.5e-19< / td >
< td class = "org-right" > -5.8e-19< / td >
< / tr >
< tr >
< td class = "org-right" > -5.7e-19< / td >
< td class = "org-right" > -0.03< / td >
< td class = "org-right" > -1.5e-17< / td >
< td class = "org-right" > 0.0085< / td >
< td class = "org-right" > 4.9e-20< / td >
< td class = "org-right" > 1.7e-19< / td >
< / tr >
< tr >
< td class = "org-right" > 0.03< / td >
< td class = "org-right" > 2.2e-19< / td >
< td class = "org-right" > 6.5e-19< / td >
< td class = "org-right" > 4.9e-20< / td >
< td class = "org-right" > 0.0085< / td >
< td class = "org-right" > -1.1e-18< / td >
< / tr >
< tr >
< td class = "org-right" > 1.7e-19< / td >
< td class = "org-right" > -5.3e-19< / td >
< td class = "org-right" > -5.8e-19< / td >
< td class = "org-right" > 1.7e-19< / td >
< td class = "org-right" > -1.1e-18< / td >
< td class = "org-right" > 0.032< / td >
< / tr >
< / tbody >
< / table >
< p >
We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiffness matrix is not diagonal.
< / p >
< / div >
< / div >
2019-03-26 09:25:04 +01:00
< div id = "outline-container-org6729b53" class = "outline-3" >
< h3 id = "org6729b53" > < span class = "section-number-3" > 2.5< / span > Conclusion< / h3 >
2019-03-25 18:12:43 +01:00
< div class = "outline-text-3" id = "text-2-5" >
< div class = "important" >
< ul class = "org-ul" >
< li > The cubic configuration permits to have \(k_x = k_y = k_z\) and \(k_{\theta\x} = k_{\theta_y}\)< / li >
< li > The stiffness matrix \(K\) is diagonal for the cubic configuration if the Stewart platform and the cube are centered < b > and< / b > the Jacobian is estimated at the cube center< / li >
< / ul >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org0969b06" class = "outline-2" >
< h2 id = "org0969b06" > < span class = "section-number-2" > 3< / span > Cubic size analysis< / h2 >
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< div class = "outline-text-2" id = "text-3" >
< p >
We here study the effect of the size of the cube used for the Stewart configuration.
< / p >
< p >
We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform.
< / p >
< p >
We only vary the size of the cube.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > H_cubes = < span style = "color: #BFEBBF;" > 250< / span > < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > 20< / span > < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > 350< / span > ;
stewarts = < span style = "color: #DCDCCC;" > {< / span > zeros< span style = "color: #BFEBBF;" > (< / span > length< span style = "color: #D0BF8F;" > (< / span > H_cubes< span style = "color: #D0BF8F;" > )< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > }< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :length< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > H_cubes< / span > < span style = "color: #DCDCCC;" > )< / span >
H_cube = H_cubes< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #DCDCCC;" > )< / span > ;
H_tot = < span style = "color: #BFEBBF;" > 100< / span > ;
H = < span style = "color: #BFEBBF;" > 80< / span > ;
opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'H_tot'< / span > , H_tot, < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Total height of the Hexapod [mm]< / span >
< span style = "color: #CC9393;" > 'L'< / span > , H_cube< span style = "color: #7CB8BB;" > /< / span > sqrt< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Size of the Cube [mm]< / span >
< span style = "color: #CC9393;" > 'H'< / span > , H, < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between base joints and platform joints [mm]< / span >
< span style = "color: #CC9393;" > 'H0'< / span > , H_cube< span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > -< / span > H< span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between the corner of the cube and the plane containing the base joints [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
stewart = initializeCubicConfiguration< span style = "color: #DCDCCC;" > (< / span > opts< span style = "color: #DCDCCC;" > )< / span > ;
opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'Jd_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , H_cube< span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > -< / span > opts.H0< span style = "color: #7CB8BB;" > -< / span > opts.H_tot< span style = "color: #BFEBBF;" > ]< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]< / span >
< span style = "color: #CC9393;" > 'Jf_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , H_cube< span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > -< / span > opts.H0< span style = "color: #7CB8BB;" > -< / span > opts.H_tot< span style = "color: #BFEBBF;" > ]< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for force location from the top of the mobile platform [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
stewart = computeGeometricalProperties< span style = "color: #DCDCCC;" > (< / span > stewart, opts< span style = "color: #DCDCCC;" > )< / span > ;
stewarts< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #DCDCCC;" > )< / span > = < span style = "color: #DCDCCC;" > {< / span > stewart< span style = "color: #DCDCCC;" > }< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< p >
The Stiffness matrix is computed for all generated Stewart platforms.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Ks = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 6< / span > , length< span style = "color: #BFEBBF;" > (< / span > H_cube< span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :length< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > H_cubes< / span > < span style = "color: #DCDCCC;" > )< / span >
Ks< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #7CB8BB;" > :< / span > , < span style = "color: #7CB8BB;" > :< / span > , < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #DCDCCC;" > )< / span > = stewarts< span style = "color: #DCDCCC;" > {< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #DCDCCC;" > }< / span > .Jd'< span style = "color: #7CB8BB;" > *< / span > stewarts< span style = "color: #DCDCCC;" > {< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #DCDCCC;" > }< / span > .Jd;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< p >
The only elements of \(K\) that vary are \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\).
< / p >
< p >
Finally, we plot \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\)
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #7CB8BB;" > figure< / span > ;
hold on;
plot< span style = "color: #DCDCCC;" > (< / span > H_cubes, squeeze< span style = "color: #BFEBBF;" > (< / span > Ks< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > 4< / span > , < span style = "color: #BFEBBF;" > 4< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #CC9393;" > 'DisplayName', '< / span > $k_< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #7CB8BB;" > \< / span > theta_x< span style = "color: #BFEBBF;" > }< / span > $'< span style = "color: #DCDCCC;" > )< / span > ;
plot< span style = "color: #DCDCCC;" > (< / span > H_cubes, squeeze< span style = "color: #BFEBBF;" > (< / span > Ks< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #CC9393;" > 'DisplayName', '< / span > $k_< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #7CB8BB;" > \< / span > theta_z< span style = "color: #BFEBBF;" > }< / span > $'< span style = "color: #DCDCCC;" > )< / span > ;
hold off;
legend< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #CC9393;" > 'location', 'northwest'< / span > < span style = "color: #DCDCCC;" > )< / span > ;
xlabel< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #CC9393;" > 'Cube Size < / span > < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #CC9393;" > mm< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "color: #CC9393;" > '< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #CC9393;" > ; ylabel< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #CC9393;" > 'Rotational stiffnes < / span > < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #CC9393;" > normalized< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "color: #CC9393;" > '< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
< / div >
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< div id = "orgad7fc21" class = "figure" >
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< p > < img src = "figs/stiffness_cube_size.png" alt = "stiffness_cube_size.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube< / p >
< / div >
< p >
We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
< / p >
< div class = "important" >
< p >
In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.
In that case, the legs will the further separated. Size of the cube is then limited by allowed space.
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org0f56064" class = "outline-2" >
< h2 id = "org0f56064" > < span class = "section-number-2" > 4< / span > initializeCubicConfiguration< / h2 >
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< div class = "outline-text-2" id = "text-4" >
< p >
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< a id = "org7e73a4b" > < / a >
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< / p >
< / div >
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< div id = "outline-container-orge3f31b8" class = "outline-3" >
< h3 id = "orge3f31b8" > < span class = "section-number-3" > 4.1< / span > Function description< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > function< / span > < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #DFAF8F;" > stewart< / span > < span style = "color: #DCDCCC;" > ]< / span > = < span style = "color: #93E0E3;" > initializeCubicConfiguration< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #DFAF8F;" > opts_param< / span > < span style = "color: #DCDCCC;" > )< / span >
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org2580fde" class = "outline-3" >
< h3 id = "org2580fde" > < span class = "section-number-3" > 4.2< / span > Optional Parameters< / h3 >
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< div class = "outline-text-3" id = "text-4-2" >
< p >
Default values for opts.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'H_tot'< / span > , < span style = "color: #BFEBBF;" > 90< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Total height of the Hexapod [mm]< / span >
< span style = "color: #CC9393;" > 'L'< / span > , < span style = "color: #BFEBBF;" > 110< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Size of the Cube [mm]< / span >
< span style = "color: #CC9393;" > 'H'< / span > , < span style = "color: #BFEBBF;" > 40< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between base joints and platform joints [mm]< / span >
< span style = "color: #CC9393;" > 'H0'< / span > , < span style = "color: #BFEBBF;" > 75< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height between the corner of the cube and the plane containing the base joints [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
< / div >
< p >
Populate opts with input parameters
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > if< / span > exist< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #CC9393;" > 'opts_param','var'< / span > < span style = "color: #DCDCCC;" > )< / span >
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > opt< / span > = < span style = "color: #BFEBBF;" > fieldnames< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > opts_param< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #BFEBBF;" > '< / span >
opts.< span style = "color: #DCDCCC;" > (< / span > opt< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > }< / span > < span style = "color: #DCDCCC;" > )< / span > = opts_param.< span style = "color: #DCDCCC;" > (< / span > opt< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > }< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org230a253" class = "outline-3" >
< h3 id = "org230a253" > < span class = "section-number-3" > 4.3< / span > Cube Creation< / h3 >
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< div class = "outline-text-3" id = "text-4-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > points = < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 1< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 1< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #DCDCCC;" > ]< / span > ;
points = opts.L< span style = "color: #7CB8BB;" > *< / span > points;
< / pre >
< / div >
< p >
We create the rotation matrix to rotate the cube
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > sx = cross< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > ]< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "color: #DCDCCC;" > )< / span > ;
sx = sx< span style = "color: #7CB8BB;" > /< / span > norm< span style = "color: #DCDCCC;" > (< / span > sx< span style = "color: #DCDCCC;" > )< / span > ;
sy = < span style = "color: #7CB8BB;" > -< / span > cross< span style = "color: #DCDCCC;" > (< / span > sx, < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "color: #DCDCCC;" > )< / span > ;
sy = sy< span style = "color: #7CB8BB;" > /< / span > norm< span style = "color: #DCDCCC;" > (< / span > sy< span style = "color: #DCDCCC;" > )< / span > ;
sz = < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #DCDCCC;" > ]< / span > ;
sz = sz< span style = "color: #7CB8BB;" > /< / span > norm< span style = "color: #DCDCCC;" > (< / span > sz< span style = "color: #DCDCCC;" > )< / span > ;
R = < span style = "color: #DCDCCC;" > [< / span > sx', sy', sz'< span style = "color: #DCDCCC;" > ]< / span > ';
< / pre >
< / div >
< p >
We use to rotation matrix to rotate the cube
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > cube = zeros< span style = "color: #DCDCCC;" > (< / span > size< span style = "color: #BFEBBF;" > (< / span > points< span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :size< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > points, < / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #DCDCCC;" > )< / span >
cube< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = R < span style = "color: #7CB8BB;" > *< / span > points< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ';
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgd1a04ef" class = "outline-3" >
< h3 id = "orgd1a04ef" > < span class = "section-number-3" > 4.4< / span > Vectors of each leg< / h3 >
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< div class = "outline-text-3" id = "text-4-4" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > leg_indices = < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 3< / span > , < span style = "color: #BFEBBF;" > 4< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 2< / span > , < span style = "color: #BFEBBF;" > 4< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 2< / span > , < span style = "color: #BFEBBF;" > 6< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 5< / span > , < span style = "color: #BFEBBF;" > 6< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 5< / span > , < span style = "color: #BFEBBF;" > 7< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 3< / span > , < span style = "color: #BFEBBF;" > 7< / span > < span style = "color: #DCDCCC;" > ]< / span > ;
< / pre >
< / div >
< p >
Vectors are:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > legs = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
legs_start = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
legs< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = cube< span style = "color: #DCDCCC;" > (< / span > leg_indices< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > -< / span > cube< span style = "color: #DCDCCC;" > (< / span > leg_indices< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ;
legs_start< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = cube< span style = "color: #DCDCCC;" > (< / span > leg_indices< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgfec0778" class = "outline-3" >
< h3 id = "orgfec0778" > < span class = "section-number-3" > 4.5< / span > Verification of Height of the Stewart Platform< / h3 >
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< div class = "outline-text-3" id = "text-4-5" >
< p >
If the Stewart platform is not contained in the cube, throw an error.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Hmax = cube< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 4< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > -< / span > cube< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 2< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > if< / span > opts.H0 < span style = "color: #7CB8BB;" > < < / span > cube< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 2< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span >
error< span style = "color: #DCDCCC;" > (< / span > sprintf< span style = "color: #BFEBBF;" > (< / span > 'H0 is not high enought. Minimum H0 = %.< span style = "color: #BFEBBF;" > 1f< / span > ', cube(< span style = "color: #BFEBBF;" > 2< / span > , < span style = "color: #BFEBBF;" > 3< / span > )));
< span style = "color: #F0DFAF; font-weight: bold;" > else< / span > < span style = "color: #F0DFAF; font-weight: bold;" > if< / span > opts.H0 < span style = "color: #7CB8BB;" > +< / span > opts.H < span style = "color: #7CB8BB;" > > < / span > cube< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > 4< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #D0BF8F;" > )< / span >
error< span style = "color: #D0BF8F;" > (< / span > sprintf< span style = "color: #93E0E3;" > (< / span > 'H0< span style = "color: #7CB8BB;" > +< / span > H is too high. Maximum H0< span style = "color: #7CB8BB;" > +< / span > H = %.< span style = "color: #BFEBBF;" > 1f< / span > ', cube(< span style = "color: #BFEBBF;" > 4< / span > , < span style = "color: #BFEBBF;" > 3< / span > )));
error< span style = "color: #9FC59F;" > (< / span > < span style = "color: #CC9393;" > 'H0+H is too high'< / span > < span style = "color: #9FC59F;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgc3cd1d1" class = "outline-3" >
< h3 id = "orgc3cd1d1" > < span class = "section-number-3" > 4.6< / span > Determinate the location of the joints< / h3 >
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< div class = "outline-text-3" id = "text-4-6" >
< p >
We now determine the location of the joints on the fixed platform w.r.t the fixed frame \(\{A\}\).
\(\{A\}\) is fixed to the bottom of the base.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Aa = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
t = < span style = "color: #DCDCCC;" > (< / span > opts.H0< span style = "color: #7CB8BB;" > -< / span > legs_start< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #DCDCCC;" > (< / span > legs< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
Aa< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = legs_start< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > +< / span > t< span style = "color: #7CB8BB;" > *< / span > legs< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< p >
And the location of the joints on the mobile platform with respect to \(\{A\}\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Ab = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
t = < span style = "color: #DCDCCC;" > (< / span > opts.H0< span style = "color: #7CB8BB;" > +< / span > opts.H< span style = "color: #7CB8BB;" > -< / span > legs_start< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #DCDCCC;" > (< / span > legs< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
Ab< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = legs_start< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > +< / span > t< span style = "color: #7CB8BB;" > *< / span > legs< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< p >
And the location of the joints on the mobile platform with respect to \(\{B\}\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Bb = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
Bb = Ab < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #DCDCCC;" > (< / span > opts.H0 < span style = "color: #7CB8BB;" > +< / span > opts.H_tot< span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > +< / span > opts.H< span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #DCDCCC;" > ]< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > h = opts.H0 < span style = "color: #7CB8BB;" > +< / span > opts.H< span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > -< / span > opts.H_tot< span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > ;
Aa = Aa < span style = "color: #7CB8BB;" > -< / span > h< span style = "color: #7CB8BB;" > *< / span > < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #DCDCCC;" > ]< / span > ;
Ab = Ab < span style = "color: #7CB8BB;" > -< / span > h< span style = "color: #7CB8BB;" > *< / span > < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #DCDCCC;" > ]< / span > ;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org9063896" class = "outline-3" >
< h3 id = "org9063896" > < span class = "section-number-3" > 4.7< / span > Returns Stewart Structure< / h3 >
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< div class = "outline-text-3" id = "text-4-7" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart = struct< span style = "color: #DCDCCC;" > ()< / span > ;
stewart.Aa = Aa;
stewart.Ab = Ab;
stewart.Bb = Bb;
stewart.H_tot = opts.H_tot;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org5807740" class = "outline-2" >
< h2 id = "org5807740" > < span class = "section-number-2" > 5< / span > Tests< / h2 >
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< div class = "outline-text-2" id = "text-5" >
< / div >
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< div id = "outline-container-org8809c47" class = "outline-3" >
< h3 id = "org8809c47" > < span class = "section-number-3" > 5.1< / span > First attempt to parametrisation< / h3 >
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< div class = "outline-text-3" id = "text-5-1" >
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< div id = "org520c5d6" class = "figure" >
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< p > < img src = "./figs/stewart_bottom_plate.png" alt = "stewart_bottom_plate.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Schematic of the bottom plates with all the parameters< / p >
< / div >
< p >
The goal is to choose \(\alpha\), \(\beta\), \(R_\text{leg, t}\) and \(R_\text{leg, b}\) in such a way that the configuration is cubic.
< / p >
< p >
The configuration is cubic if:
\[ \overrightarrow{a_i b_i} \cdot \overrightarrow{a_j b_j} = 0, \ \forall i, j = [1, \hdots, 6], i \ne j \]
< / p >
< p >
Lets express \(a_i\), \(b_i\) and \(a_j\):
< / p >
\begin{equation*}
a_1 = \begin{bmatrix}R_{\text{leg,b}} \cos(120 - \alpha) \\ R_{\text{leg,b}} \cos(120 - \alpha) \\ 0\end{bmatrix} ; \quad
a_2 = \begin{bmatrix}R_{\text{leg,b}} \cos(120 + \alpha) \\ R_{\text{leg,b}} \cos(120 + \alpha) \\ 0\end{bmatrix} ; \quad
\end{equation*}
\begin{equation*}
b_1 = \begin{bmatrix}R_{\text{leg,t}} \cos(120 - \beta) \\ R_{\text{leg,t}} \cos(120 - \beta\\ H\end{bmatrix} ; \quad
b_2 = \begin{bmatrix}R_{\text{leg,t}} \cos(120 + \beta) \\ R_{\text{leg,t}} \cos(120 + \beta\\ H\end{bmatrix} ; \quad
\end{equation*}
< p >
\[ \overrightarrow{a_1 b_1} = b_1 - a_1 = \begin{bmatrix}R_{\text{leg}} \cos(120 - \alpha) \\ R_{\text{leg}} \cos(120 - \alpha) \\ 0\end{bmatrix}\]
< / p >
< / div >
< / div >
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< div id = "outline-container-org71c64bc" class = "outline-3" >
< h3 id = "org71c64bc" > < span class = "section-number-3" > 5.2< / span > Second attempt< / h3 >
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< div class = "outline-text-3" id = "text-5-2" >
< p >
We start with the point of a cube in space:
< / p >
\begin{align*}
[0, 0, 0] ; \ [0, 0, 1]; \ ...
\end{align*}
< p >
We also want the cube to point upward:
\[ [1, 1, 1] \Rightarrow [0, 0, 1] \]
< / p >
< p >
Then we have the direction of all the vectors expressed in the frame of the hexapod.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > points = < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 1< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 1< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #DCDCCC;" > ]< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #7CB8BB;" > figure< / span > ;
plot3< span style = "color: #DCDCCC;" > (< / span > points< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #7CB8BB;" > :< / span > ,< span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > )< / span > , points< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #7CB8BB;" > :< / span > ,< span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #BFEBBF;" > )< / span > , points< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #7CB8BB;" > :< / span > ,< span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #CC9393;" > 'ko'< / span > < span style = "color: #DCDCCC;" > )< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > sx = cross< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > ]< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "color: #DCDCCC;" > )< / span > ;
sx = sx< span style = "color: #7CB8BB;" > /< / span > norm< span style = "color: #DCDCCC;" > (< / span > sx< span style = "color: #DCDCCC;" > )< / span > ;
sy = < span style = "color: #7CB8BB;" > -< / span > cross< span style = "color: #DCDCCC;" > (< / span > sx, < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "color: #DCDCCC;" > )< / span > ;
sy = sy< span style = "color: #7CB8BB;" > /< / span > norm< span style = "color: #DCDCCC;" > (< / span > sy< span style = "color: #DCDCCC;" > )< / span > ;
sz = < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #DCDCCC;" > ]< / span > ;
sz = sz< span style = "color: #7CB8BB;" > /< / span > norm< span style = "color: #DCDCCC;" > (< / span > sz< span style = "color: #DCDCCC;" > )< / span > ;
R = < span style = "color: #DCDCCC;" > [< / span > sx', sy', sz'< span style = "color: #DCDCCC;" > ]< / span > ';
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > cube = zeros< span style = "color: #DCDCCC;" > (< / span > size< span style = "color: #BFEBBF;" > (< / span > points< span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :size< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > points, < / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #DCDCCC;" > )< / span >
cube< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = R < span style = "color: #7CB8BB;" > *< / span > points< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ';
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #7CB8BB;" > figure< / span > ;
hold on;
plot3< span style = "color: #DCDCCC;" > (< / span > points< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #7CB8BB;" > :< / span > ,< span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > )< / span > , points< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #7CB8BB;" > :< / span > ,< span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #BFEBBF;" > )< / span > , points< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #7CB8BB;" > :< / span > ,< span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #CC9393;" > 'ko'< / span > < span style = "color: #DCDCCC;" > )< / span > ;
plot3< span style = "color: #DCDCCC;" > (< / span > cube< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #7CB8BB;" > :< / span > ,< span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > )< / span > , cube< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #7CB8BB;" > :< / span > ,< span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #BFEBBF;" > )< / span > , cube< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #7CB8BB;" > :< / span > ,< span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #CC9393;" > 'ro'< / span > < span style = "color: #DCDCCC;" > )< / span > ;
hold off;
< / pre >
< / div >
< p >
Now we plot the legs of the hexapod.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > leg_indices = < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 3< / span > , < span style = "color: #BFEBBF;" > 4< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 2< / span > , < span style = "color: #BFEBBF;" > 4< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 2< / span > , < span style = "color: #BFEBBF;" > 6< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 5< / span > , < span style = "color: #BFEBBF;" > 6< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 5< / span > , < span style = "color: #BFEBBF;" > 7< / span > ; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 3< / span > , < span style = "color: #BFEBBF;" > 7< / span > < span style = "color: #DCDCCC;" > ]< / span >
< span style = "color: #7CB8BB;" > figure< / span > ;
hold on;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
plot3< span style = "color: #DCDCCC;" > (< / span > cube< span style = "color: #BFEBBF;" > (< / span > leg_indices< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #D0BF8F;" > )< / span > ,< span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > )< / span > , cube< span style = "color: #BFEBBF;" > (< / span > leg_indices< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #D0BF8F;" > )< / span > ,< span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #BFEBBF;" > )< / span > , cube< span style = "color: #BFEBBF;" > (< / span > leg_indices< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #D0BF8F;" > )< / span > ,< span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #CC9393;" > '-'< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
hold off;
< / pre >
< / div >
< p >
Vectors are:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > legs = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
legs_start = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
legs< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = cube< span style = "color: #DCDCCC;" > (< / span > leg_indices< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > -< / span > cube< span style = "color: #DCDCCC;" > (< / span > leg_indices< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ;
legs_start< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = cube< span style = "color: #DCDCCC;" > (< / span > leg_indices< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span >
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< p >
We now have the orientation of each leg.
< / p >
< p >
We here want to see if the position of the "slice" changes something.
< / p >
< p >
Let's first estimate the maximum height of the Stewart platform.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Hmax = cube< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 4< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > -< / span > cube< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 2< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
< / div >
< p >
Let's then estimate the middle position of the platform
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Hmid = cube< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 8< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > ;
< / pre >
< / div >
< / div >
< / div >
2019-03-26 09:25:04 +01:00
< div id = "outline-container-orgc132128" class = "outline-3" >
< h3 id = "orgc132128" > < span class = "section-number-3" > 5.3< / span > Generate the Stewart platform for a Cubic configuration< / h3 >
2019-03-25 18:12:43 +01:00
< div class = "outline-text-3" id = "text-5-3" >
< p >
First we defined the height of the Hexapod.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > H = Hmax< span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 2< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Zs = < span style = "color: #BFEBBF;" > 1< / span > .< span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > *< / span > cube< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 2< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ; < span style = "color: #7F9F7F;" > % Height of the fixed platform< / span >
Ze = Zs < span style = "color: #7CB8BB;" > +< / span > H; < span style = "color: #7F9F7F;" > % Height of the mobile platform< / span >
< / pre >
< / div >
< p >
We now determine the location of the joints on the fixed platform.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Aa = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
t = < span style = "color: #DCDCCC;" > (< / span > Zs< span style = "color: #7CB8BB;" > -< / span > legs_start< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #DCDCCC;" > (< / span > legs< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
Aa< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = legs_start< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > +< / span > t< span style = "color: #7CB8BB;" > *< / span > legs< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< p >
And the location of the joints on the mobile platform
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Ab = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
t = < span style = "color: #DCDCCC;" > (< / span > Ze< span style = "color: #7CB8BB;" > -< / span > legs_start< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #DCDCCC;" > (< / span > legs< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
Ab< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = legs_start< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > +< / span > t< span style = "color: #7CB8BB;" > *< / span > legs< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< p >
And we plot the legs.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #7CB8BB;" > figure< / span > ;
hold on;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
plot3< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > [< / span > Ab< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #D0BF8F;" > )< / span > ,Aa< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > ]< / span > , < span style = "color: #BFEBBF;" > [< / span > Ab< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #D0BF8F;" > )< / span > ,Aa< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > ]< / span > , < span style = "color: #BFEBBF;" > [< / span > Ab< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #D0BF8F;" > )< / span > ,Aa< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > ]< / span > , < span style = "color: #CC9393;" > 'k-'< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
hold off;
xlim< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "color: #DCDCCC;" > )< / span > ;
ylim< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "color: #DCDCCC;" > )< / span > ;
zlim< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
< / div >
< / div >
< / div >
< / div >
2019-03-26 09:25:04 +01:00
< p >
< h1 class = 'org-ref-bib-h1' > Bibliography< / h1 >
< ul class = 'org-ref-bib' > < li > < a id = "geng94_six_degree_of_freed_activ" > [geng94_six_degree_of_freed_activ]< / a > < a name = "geng94_six_degree_of_freed_activ" > < / a > Geng & Haynes, Six Degree-Of-Freedom Active Vibration Control Using the Stewart Platforms, < i > IEEE Transactions on Control Systems Technology< / i > , < b > 2(1)< / b > , 45-53 (1994). < a href = "https://doi.org/10.1109/87.273110" > link< / a > . < a href = "http://dx.doi.org/10.1109/87.273110" > doi< / a > .< / li >
< li > < a id = "jafari03_orthog_gough_stewar_platf_microm" > [jafari03_orthog_gough_stewar_platf_microm]< / a > < a name = "jafari03_orthog_gough_stewar_platf_microm" > < / a > Jafari & McInroy, Orthogonal Gough-Stewart Platforms for Micromanipulation, < i > IEEE Transactions on Robotics and Automation< / i > , < b > 19(4)< / b > , 595-603 (2003). < a href = "https://doi.org/10.1109/tra.2003.814506" > link< / a > . < a href = "http://dx.doi.org/10.1109/tra.2003.814506" > doi< / a > .< / li >
< li > < a id = "preumont07_six_axis_singl_stage_activ" > [preumont07_six_axis_singl_stage_activ]< / a > < a name = "preumont07_six_axis_singl_stage_activ" > < / a > Preumont, Horodinca, Romanescu, de, Marneffe, Avraam, Deraemaeker, Bossens, & Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform, < i > Journal of Sound and Vibration< / i > , < b > 300(3-5)< / b > , 644-661 (2007). < a href = "https://doi.org/10.1016/j.jsv.2006.07.050" > link< / a > . < a href = "http://dx.doi.org/10.1016/j.jsv.2006.07.050" > doi< / a > .< / li >
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< p class = "author" > Author: Thomas Dehaeze< / p >
2019-03-26 09:25:04 +01:00
< p class = "date" > Created: 2019-03-26 mar. 08:47< / p >
2019-03-25 18:12:43 +01:00
< p class = "validation" > < a href = "http://validator.w3.org/check?uri=referer" > Validate< / a > < / p >
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